On support varieties for modules over complete intersections

Bergh, Petter

doi:10.1090/S0002-9939-07-09009-0

On support varieties for modules over complete intersections
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by Petter Andreas Bergh

Proc. Amer. Math. Soc. 135 (2007), 3795-3803

DOI: https://doi.org/10.1090/S0002-9939-07-09009-0

Published electronically: September 7, 2007

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Abstract:

Let $(A, \mathfrak {m}, k)$ be a complete intersection of codimension $c$, and let $\tilde {k}$ be the algebraic closure of $k$. We show that every homogeneous algebraic subset of $\tilde {k}^c$ is the cohomological support variety of an $A$-module, and that the projective variety of a complete indecomposable maximal Cohen–Macaulay $A$-module is connected.

References

Maurice Auslander and Ragnar-Olaf Buchweitz, The homological theory of maximal Cohen-Macaulay approximations, Mém. Soc. Math. France (N.S.) 38 (1989), 5–37 (English, with French summary). Colloque en l’honneur de Pierre Samuel (Orsay, 1987). MR 1044344
L. L. Avramov, Modules of finite virtual projective dimension, Invent. Math. 96 (1989), no. 1, 71–101. MR 981738, DOI 10.1007/BF01393971
Luchezar L. Avramov and Ragnar-Olaf Buchweitz, Support varieties and cohomology over complete intersections, Invent. Math. 142 (2000), no. 2, 285–318. MR 1794064, DOI 10.1007/s002220000090
Luchezar L. Avramov, Vesselin N. Gasharov, and Irena V. Peeva, Complete intersection dimension, Inst. Hautes Études Sci. Publ. Math. 86 (1997), 67–114 (1998). MR 1608565, DOI 10.1007/BF02698901
Tokuji Araya and Yuji Yoshino, Remarks on a depth formula, a grade inequality and a conjecture of Auslander, Comm. Algebra 26 (1998), no. 11, 3793–3806. MR 1647079, DOI 10.1080/00927879808826375
P.A. Bergh, Modules with reducible complexity, J. Algebra 310 (2007), no. 1, 132–147.
Jon F. Carlson, The variety of an indecomposable module is connected, Invent. Math. 77 (1984), no. 2, 291–299. MR 752822, DOI 10.1007/BF01388448
Karin Erdmann, Miles Holloway, Rachel Taillefer, Nicole Snashall, and Øyvind Solberg, Support varieties for selfinjective algebras, $K$-Theory 33 (2004), no. 1, 67–87. MR 2199789, DOI 10.1007/s10977-004-0838-7
Alex Martsinkovsky, Cohen-Macaulay modules and approximations, Infinite length modules (Bielefeld, 1998) Trends Math., Birkhäuser, Basel, 2000, pp. 167–192. MR 1789215
Hideyuki Matsumura, Commutative ring theory, Cambridge Studies in Advanced Mathematics, vol. 8, Cambridge University Press, Cambridge, 1986. Translated from the Japanese by M. Reid. MR 879273
Nicole Snashall and Øyvind Solberg, Support varieties and Hochschild cohomology rings, Proc. London Math. Soc. (3) 88 (2004), no. 3, 705–732. MR 2044054, DOI 10.1112/S002461150301459X